The geometry at infinity of LW-spacelike submanifolds in semi-Riemannian space forms
Volume 175 / 2024
Abstract
We deal with complete noncompact linear Weingarten spacelike submanifolds (for short, LW-spacelike submanifolds) immersed with parallel normalized mean curvature vector into semi-Riemannian space forms $\mathbb Q_{p}^{n+p}(c)$ of index $p$ and constant sectional curvature $c$. Under suitable restrictions on the behavior of the mean curvature at infinity and values of the norm of the traceless part of the second fundamental form, we show that such an LW-spacelike submanifold must be isometric to one of the following hyperbolic cylinders: $\mathbb R^{n-1}\times \mathbb H^1(c_2)$, with $c_2 \lt 0$, when $c=0$; $\mathbb S^{n-1}(c_1)\times \mathbb H^1(c_2)$, with $c_1 \gt 0$, $c_2 \lt 0$ and $1/c_1+1/c_2=1/c$, when $c \gt 0$; $\mathbb H^{n-1}(c_1)\times \mathbb H^1(c_2)$, with $c_1 \lt 0$, $c_2 \lt 0$ and $1/c_1+1/c_2=1/c$, when $c \lt 0$. Our approach is based on the use of a new form of maximum principle on complete noncompact Riemannian manifolds, due to Alías, Caminha and Nascimento (2019).
